Chapter 3.3, Problem 29PPSE

Solution

To solve: The system of inequality using the graphing method $\left(\begin{array}{l}-x\ge 4-y\\ y\ge \left|3x-6\right|\end{array}\right.$ .

The overlapping region is the solution region.

Given information:

The given system of inequalities is,

  $\left(\begin{array}{l}-x\ge 4-y\\ y\ge \left|3x-6\right|\end{array}\right.$ .

Concept used:

Graph each inequality and then overlapping region is the solution of the system.

Plot the graph:

Consider the inequality $-x\ge 4-y$.

Rewrite the inequality as $4-y\le -x$

Subtract both side $-4$ and multiply both side by $-1$ to get to get in slope form , then inequality becomes $y\le x+4$ .

The graph of the inequality $y\le x+4$ is given below.

  

The graph of the inequality $y\ge \left|3x-6\right|$ is given below:

  

The graph of the system of inequalities $\left(\begin{array}{l}-x\ge 4-y\\ y\ge \left|3x-6\right|\end{array}\right.$ is given below:

  

Now to check any two point from the overlapping region and check the solution region is true or not. Take the point $\left(2,1\right)$

For $\left(2,1\right)$ check $y\le x+4\text{and}\text{y}\ge \left|3x-6\right|$ :

  $2<5,1>0$ ,both inequalities are true to the point $\left(3,-1\right)$ .

So the overlapping region is the solution of the system of given inequalities.